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Finding planar surfaces in knot- and link-manifolds. (English) Zbl 1176.57024

A link-manifold is a compact orientable 3-manifold whose boundary consists of tori. In particular, if the boundary is a single torus, then the manifold is called a knot-manifold. The main result of the paper under review claims that given any link-manifold, there is an algorithm to decide whether the manifold contains an embedded essential planar surface. Moreover, if it does, then the algorithm constructs such surface.
Also, two further results, with boundary conditions, are presented. Given a link-manifold \(M\), a boundary component \(B\), and a slope \(\gamma\) in \(B\), there is an algorithm to decide whether \(M\) contains an embedded punctured disk with boundary having slope \(\gamma\) and punctures in \(\partial M-B\). If it does, again the algorithm constructs one. Another result gives an algorithm to decide if \(M\) contains such a punctured disk with boundary having slope of a longitude, which means a slope meeting the given slope \(\gamma\) in one point.
The arguments are based on normal surface theory, including a number of new tools, and new results on triangulations. An interesting aspect is that the algorithms developed in the paper do not necessarily find an answer among the fundamental surfaces. The last section discusses the word problem for 3-manifold groups.

MSC:

57N10 Topology of general \(3\)-manifolds (MSC2010)
57M99 General low-dimensional topology
57M50 General geometric structures on low-dimensional manifolds
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